Jacobi elliptic

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11: 22.15 Inverse Functions

… ►are denoted respectively by ►

$\xi =\mathrm{arcsn}(x,k),$

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$\eta =\mathrm{arccn}(x,k),$

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$\zeta =\mathrm{arcdn}(x,k).$

… ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to …

12: 22.5 Special Values

… ►Table 22.5.2 gives $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$ for other special values of $z$. … ►

Table 22.5.3: Limiting forms of Jacobian elliptic functions as $k\to 0$.

► ►►►►

$\mathrm{sn}(z,k)$ $\to$	$\sin z$	$\mathrm{cd}(z,k)$ $\to$	$\cos z$	$\mathrm{dc}(z,k)$ $\to$	$\sec z$	$\mathrm{ns}(z,k)$ $\to$	$\csc z$
$\mathrm{cn}(z,k)$ $\to$	$\cos z$	$\mathrm{sd}(z,k)$ $\to$	$\sin z$	$\mathrm{nc}(z,k)$ $\to$	$\sec z$	$\mathrm{ds}(z,k)$ $\to$	$\csc z$
$\mathrm{dn}(z,k)$ $\to$	$1$	$\mathrm{nd}(z,k)$ $\to$	$1$	$\mathrm{sc}(z,k)$ $\to$	$\tan z$	$\mathrm{cs}(z,k)$ $\to$	$\cot z$

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Table 22.5.4: Limiting forms of Jacobian elliptic functions as $k\to 1$.

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$\mathrm{sn}(z,k)$ $\to$	$\tanh z$	$\mathrm{cd}(z,k)$ $\to$	$1$	$\mathrm{dc}(z,k)$ $\to$	$1$	$\mathrm{ns}(z,k)$ $\to$	$\coth z$
…

►

…

13: 22.3 Graphics

… ►Line graphs of the functions $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $\mathrm{cd}(x,k)$, $\mathrm{sd}(x,k)$, $\mathrm{nd}(x,k)$, $\mathrm{dc}(x,k)$, $\mathrm{nc}(x,k)$, $\mathrm{sc}(x,k)$, $\mathrm{ns}(x,k)$, $\mathrm{ds}(x,k)$, and $\mathrm{cs}(x,k)$ for representative values of real $x$ and real $k$ illustrating the near trigonometric ($k=0$), and near hyperbolic ($k=1$) limits. … ► $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, and $\mathrm{dn}(x,k)$ as functions of real arguments $x$ and $k$. … ►

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Figure 22.3.13: $\mathrm{sn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.14: $\mathrm{cn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.15: $\mathrm{dn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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14: 22.2 Definitions

… ►

22.2.4 $$\mathrm{sn}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_2(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_4(\zeta ,q)}=\frac{1}{\mathrm{ns}(z,k)},$$

ⓘ

Defines:

$\mathrm{ns}(z,k)$: Jacobian elliptic function and $\mathrm{sn}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

►

22.2.5 $$\mathrm{cn}(z,k)=\frac{{\theta }_4(0,q)}{{\theta }_2(0,q)}\frac{{\theta }_2(\zeta ,q)}{{\theta }_4(\zeta ,q)}=\frac{1}{\mathrm{nc}(z,k)},$$

ⓘ

Defines:

$\mathrm{cn}(z,k)$: Jacobian elliptic function and $\mathrm{nc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►

22.2.7 $$\mathrm{sd}(z,k)=\frac{{\theta }_3^2(0,q)}{{\theta }_2(0,q){\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_3(\zeta ,q)}=\frac{1}{\mathrm{ds}(z,k)},$$

ⓘ

Defines:

$\mathrm{ds}(z,k)$: Jacobian elliptic function and $\mathrm{sd}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/22.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

… ►

22.2.10 $$\mathrm{p}\mathrm{q}(z,k)=\frac{\mathrm{p}\mathrm{r}(z,k)}{\mathrm{q}\mathrm{r}(z,k)}=\frac{1}{\mathrm{q}\mathrm{p}(z,k)},$$

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Defines:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function

Symbols:

$z$: complex and $k$: modulus

A&S Ref:

16.3.4

Referenced by:

§22.14(iv), §22.20(ii), §22.4(ii), §22.5(i)

Permalink:

http://dlmf.nist.gov/22.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2, §22.2 and Ch.22

… ► $\mathrm{s}\mathrm{s}(z,k)=1$. …

15: 22.10 Maclaurin Series

… ►

22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

►

22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

… ►

22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

►

22.10.6 $$\mathrm{dn}(z,k)=1-\frac{k^2}{2}{\sin }^{2}z+O\left(k^4\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.3

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

…

16: 29.18 Mathematical Applications

… ►

$x=kr\mathrm{sn}(\beta ,k)\mathrm{sn}(\gamma ,k),$

►

$y=\mathrm{i}{\displaystyle \frac{k}{k^{\prime }}}r\mathrm{cn}(\beta ,k)\mathrm{cn}(\gamma ,k),$

… ►

$x=k\mathrm{sn}(\alpha ,k)\mathrm{sn}(\beta ,k)\mathrm{sn}(\gamma ,k),$

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$y=-{\displaystyle \frac{k}{k^{\prime }}}\mathrm{cn}(\alpha ,k)\mathrm{cn}(\beta ,k)\mathrm{cn}(\gamma ,k),$

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$z={\displaystyle \frac{\mathrm{i}}{k{k}^{\prime }}}\mathrm{dn}(\alpha ,k)\mathrm{dn}(\beta ,k)\mathrm{dn}(\gamma ,k),$

…

17: 29.11 Lamé Wave Equation

… ►

29.11.1 $$\frac{d^{2}w}{{dz}^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(z,k)+k^2{\omega }^2{\mathrm{sn}}^4(z,k))w=0,$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\omega$: parameter and $w(z)$: solution

Referenced by:

§29.11, §29.11, §29.18(ii), §29.7(ii)

Permalink:

http://dlmf.nist.gov/29.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.11 and Ch.29

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18: 29.8 Integral Equations

… ►

29.8.1 $$x=k^2\mathrm{sn}(z,k)\mathrm{sn}(z_1,k)\mathrm{sn}(z_2,k)\mathrm{sn}(z_3,k)-\frac{k^2}{k_{}^{\prime }{}_{}{}^2}\mathrm{cn}(z,k)\mathrm{cn}(z_1,k)\mathrm{cn}(z_2,k)\mathrm{cn}(z_3,k)+\frac{1}{k_{}^{\prime }{}_{}{}^2}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k)\mathrm{dn}(z_2,k)\mathrm{dn}(z_3,k),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $x$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

►where $z,z_1,z_2,z_3$ are real, and $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ are the Jacobian elliptic functions (§22.2). … ►

29.8.6 $$y=\frac{1}{k^{\prime }}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $y$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

19: 22.18 Mathematical Applications

… ►

$x=a\mathrm{sn}(u,k),$

►

$y=b\mathrm{cn}(u,k),$

… ►

22.18.5 $$r=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2}),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

… ►By use of the functions $\mathrm{sn}$ and $\mathrm{cn}$, parametrizations of algebraic equations, such as … ►With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

20: 22.11 Fourier and Hyperbolic Series

… ►

22.11.1 $$\mathrm{sn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.1

Permalink:

http://dlmf.nist.gov/22.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.2 $$\mathrm{cn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.2

Permalink:

http://dlmf.nist.gov/22.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

… ►Similar expansions for ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ follow immediately from (22.6.1). … ►Again, similar expansions for ${\mathrm{cn}}^2(z,k)$ and ${\mathrm{dn}}^2(z,k)$ may be derived via (22.6.1). …